By Eric Lehman, F Thomson Leighton, Albert R Meyer

This e-book covers trouble-free discrete arithmetic for computing device technology and engineering. It emphasizes mathematical definitions and proofs in addition to acceptable tools. subject matters comprise formal good judgment notation, facts tools; induction, well-ordering; units, kin; effortless graph concept; integer congruences; asymptotic notation and development of features; variations and combos, counting ideas; discrete chance. additional chosen issues can also be coated, comparable to recursive definition and structural induction; nation machines and invariants; recurrences; producing functions.

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So, the Well Ordering Principle captures something special about the nonnegative integers. 1 Well Ordering Proofs While the Well Ordering Principle may seem obvious, it’s hard to see offhand why it is useful. But in fact, it provides one of the most important proof rules in discrete mathematics. Inp fact, looking back, we took the Well Ordering Principle for granted in proving that 2 is irrational. That proof assumed that for any positive integers m and n, the fraction m=n can be written in lowest terms, that is, in the form m0 =n0 where m0 and n0 are positive integers with no common factors.

1P stands for problems whose instances can be solved in time that grows polynomially with the size of the instance. NP stands for nondeterministic polynomial time, but we’ll leave an explanation of what that is to texts on the theory of computational complexity. 6. x/ is true. x/ is true for every x in the set, D. For all x 2 R, x 2 0. x 2 0 for every x 2 R. x/ is true. x/ is true for some x in the set, D. x/ is true for at least one x 2 D. There is an x 2 R such that 5x 2 7 D 0. 5x 2 7 D 0 for some x 2 R.

2 Mixing Quantifiers Many mathematical statements involve several quantifiers. 7: Every even integer greater than 2 is the sum of two primes. “mcs” — 2011/6/2 — 10:18 — page 52 — #60 52 Chapter 3 Logical Formulas Let’s write this out in more detail to be precise about the quantification: For every even integer n greater than 2, there exist primes p and q such that n D p C q. Let Evens be the set of even integers greater than 2, and let Primes be the set of primes. 3 there exist primes p and q such that Order of Quantifiers Swapping the order of different kinds of quantifiers (existential or universal) usually changes the meaning of a proposition.